TherealorthogonalgroupandVahlenmatrices

Abstract Weprovethedifferentiabilityofagrouphomomorphism

€

ξ :O(4)→PSp(2) fromthereal orthogonal group into the projective symplectic group where

€

ξ wasconstructedin(CanluboandReyes2012).Wedescribehigherdimensionalanaloguesof

€

ξ.For

€

n ≥ 2,we consider a stereographicprojection

€

Π : Sn−1 → ˆ R n−1 from theunit sphere onto

€

ˆ R n−1 = Rn−1 ∪{∞}. ApplyingMöbius transformations and ,we embed the real orthogonalgroup intoaprojectivesubgroup ofVahlenmatriceswhere istheClifford

groupoftheCliffordalgebra ofdimension . 2000MathematicSubjectClassification:51B10,15B33 Keywords:Quaternion,Möbiustransformation,Vahlenmatrix 1Introduction Many students of mathematics are introduced to Möbius geometry by studying the linear

fractional transformation

€

FA (x) = (ax + b)(cx + d)−1

where

€

x belongstotheextendedcomplexplane

€

ˆ C = C∪{∞},and

€

A =a bc d⎛

⎝ ⎜

⎞

⎠ ⎟

has complex entries satisfying

€

ad − bc =1. The Möbius transformation

€

FA is orientationpreserving,andisaproductofanevennumberofinversionsaboutcirclesorreflectionsaboutlines(seepage57ofBeardon1983).Thefieldofcomplexnumbersmaybeenlargedtotherealalgebraofquaternions,andthequaternionsmaybefurtherenlargedtoarealCliffordalgebra,andsoon.TherearenaturalEuclideansubspacesthatsit insidetherealCliffordalgebras,seeSection 6 below. We adopt the same notation

€

FA to denote pseudo linear fractionaltransformations that are defined on the extended Euclidean subspaces of these real Cliffordalgebras.TherealCliffordalgebrashaveapplicationstothestudyofMöbius transformations on Euclidean spaces (Ahlfors 1985; Waterman 1993).

Let

€

n ≥ 2 be an integer. Let

€

Rn be the n-dimensional Euclidean space. Let

€

ˆ R n = Rn ∪{∞}betheextendedEuclideanspace.Let denotethegroupgeneratedbyinversions about (n-1)-dimensional spheres, and reflections about hyperplanes in

€

ˆ R n, seeSection5belowformoredetails.Theelementsof arecalledMöbius transformationsof ,and isthegroupofMöbius transformationsof .

Let betheunitspherecenteredattheorigin,andlet

denotethesubgroupofMöbiustransformationsof thatleavethesphere invariant.Let be a stereographic projection from the sphere onto

€

ˆ R n−1 that satisfies

€

Π(x1,...,xn ) = (1− xn )−1 x1,...,xn−1( ) if

€

xn ≠1.Weconsidertheinducedgrouphomomorphism

€

Π∗ :GM(Sn−1) →GM( ˆ R n−1) given by

€

Π∗( f ) =Π ! f !Π−1 for each . Notice that the real orthogonal group

€

O(n) isasubgroupof .Weanalyzetherestrictionof

€

Π∗ to

€

O(n) .Thespecialcase isanalyzedin(CanluboandReyes2012).Weidentify

€

R4 withthespace of real quaternions such that

€

(x0,x1,x2,x3)∈R4 is paired with the real quaternion

€

x0 + x1i+ x2 j+ x3k .Thesubspace

€

R3 is identifiedwiththespaceofpurequaternions,andtheunit sphere is the group of unit quaternions. For the stereographic projection, let

€

Π(x0,x1,x2,x3) = (1− x3)−1 x0,x1,x2( ) if

€

x3 ≠1. In (Canlubo and Reyes 2012), if

€

f ∈O(4), it isshownthat

€

Π∗( f ) = FA where

€

FA (thesamenotationusedearlierbutinadifferentcontext)isaquaternionic linear fractional transformation that isdefinedon theextendedspaceofpurequaternions,and

€

A issomematrixinthesymplecticgroup

€

Sp(2) .Furthermore,let

€

PSp(2) = Sp(2) /{±I2}.Let

€

[A]bethecosetof

€

A∈ Sp(2)in

€

PSp(2).In(CanluboandReyes2012), theauthorsprovethatthemapping

€

ξ :O(4)→PSp(2) definedby

€

ξ( f ) = [A] isawell-defined injectivegrouphomomorphism,seedetails inSection2.Thefirstmain result of this paper shows that is a differentiablemapping, see Section 3. Also, weevaluate a matrix representation for the differential

€

dξ of at the tangent space of theidentityorLiealgebraof ,seeSection4.Let

€

n ≥ 2. We describe higher dimensional analogues of the homomorphism

€

ξ :O(4)→PSp(2) . The framework forour calculations is theCliffordalgebra which isgenerated by the usual basis of such that and if

€

i ≠ j . Thefieldofcomplexnumbersis

€

Cl(1) ,andthealgebraofrealquaternionsis

€

Cl(2) .Weidentify withasubspaceof havingtheusualbasis .Let be

thesubspaceof thatisspannedby1,and .Let bethemultiplicativegroup

generatedbythenonzerovectorsin .Weconsider2-by-2matrices

€

A whereitsnonzeroentries lie in , and theentriesare subject toadditional restrictions that includeanonzero

realpseudodeterminant for

€

A , seedetails inSection6.Thesematrices

€

A arecalledVahlenmatrices, and they form a multiplicative group

€

GL2(Γn ) (see Ahlfors 1985, and Waterman1993). Let

€

SL2(Γn ) be the subgroupof

€

GL2(Γn ) that consists of Vahlenmatriceswhere thepseudo determinants are all equal to one. If

€

n =1,

€

GL2(Γ1) consists of 2-by-2 nonsingularcomplexmatriceswhosedeterminantsarereal.

We apply the Poincaré extension of a Möbius transformation

€

φ of

€

ˆ R n (page 34 ofBeardon 1983). The Poincaré extension

€

φ ext is a Möbius transformation of

€

ˆ R n+1 where therestrictionof

€

φ ext to

€

ˆ R n is

€

φ .Inconnectionstootherfields,theupperhalf-spaceof

€

Rn+1where

€

xn+1 > 0isamodelforhyperbolicgeometry,and

€

φ ext isanisometryofthisgeometry(Beardon

1985). In this paper, we realize

€

φ ext as a Möbius transformation of

€

ˆ V n+1 =V n+1 ∪{∞}, and

wheretherestrictionof

€

φ ext to

€

ˆ R n is

€

φ .We recall the stereographic projection

€

Π(x1,...,xn ) = (1− xn )−1 x1,...,xn−1( ), xn ≠1. We

showthatif

€

f ∈O(n),thentheMöbiustransformation

€

Π∗( f )of

€

ˆ R n−1mayberewrittenasapseudo linear fractional transformation

€

Π∗( f ) = FA for some

€

A∈ SL2(Γn−1) , see discussionbeforeTheorem11inSection7.Let

€

PSL2(Γn−1) = SL2(Γn−1) /{±I2},andlet

€

[A]denotethecosetin

€

PSL2(Γn−1) thatisdefinedby

€

A .Wedescribeoursecond main result. We show that there is awell-defined injective group homomorphism

€

ξen :O(n)→PSL2(Γn−1) satisfying

€

ξen ( f ) = [A](Theorem11).

Themapping

€

ξen isananalogueinhigherdimensions(butnotageneralization)ofthemapping thatwasobtainedinthequaternioniccasein(CanluboandReyes2012).Thereasonthatwedonothaveageneralization isbecausecertainquaternionicmatricesthatdefinethequaternionic linear fractional transformations are not Vahlen matrices, see details after theproofofTheorem11.

In Section8,weevaluate the left and righteigenvaluesof someVahlenmatrices. Forrelated work onMöbius geometry, and Clifford algebras see (Beardon 1983; Lawson 2010).General references for quaternions are (Coxeter 1946; Koecher and Remmert 1991; ConwayandSmith2003)

2Quaternionsandrigidmotions Let bethealgebraofquaternions ,where and

. The conjugate of is

€

x = x0 − x1i− x2 j− x3k , and the norm of is. One checks that and . Thus, the norm is

multiplicative, that is, for every . If , then we say that is a unit

quaternion.Wedenotethesetofunitquaternionsby .If ,then .Wewrite

€

Re(x) = x0fortherealpartof .If ,thenwesaythat isapurequaternion.Welet be the real space of all pure quaternions. If , then one checks that

. Notice that and are isomorphic real vector spaces. The usual

inner-productin is

€

Re(xy ) = x0y0 + x1y1 + x2y2 + x3y3. (1)Noticethat

€

xy + yx = 2Re(xy ) . A rigid motion on is a bijection satisfying for all

.Rigidmotionsof havebeencharacterized(Coxeter1946).Let bearigidmotionon .If ,then isareallinearmapping(thatis,forevery andevery ,we have ). Let be the group of rigidmotions of that fix theorigin(that is, ,andevery isalsocalledanorthogonal linearmap).Then isisomorphic to the real orthogonal matrix group . Let be given. There exist

suchthateither or .Let andlet bearigidmotion.If, then maybeexpressedas the composition of a rigidmotion that

fixestheoriginandthetranslation

€

A(x) = x + c .Let

€

y ∈S3 begiven.Foreach , let . Then , isa reallinearmappingon ,and .Let begiven.Noticethat

€

py + yp = 2Re( py ) = 0ifandonly if . Thus, if

€

Re(py ) = 0, then . In this case,we say that is areflection in about thehyperplaneperpendicular to .Onechecks that

€

fy fy x( )( ) = x forevery ,thatis, isitsowninverse.Also,wemayrewritethereflection asfollows:

€

fy (x) = x − 2Re(xy )y.Thefollowing(cf.Theorem9.1ofCoxeter1946)characterizestherigidmotionsof

thatfixtheoriginandareproductsoftwoorfourreflections.

Theorem1Let

€

p, q∈S3 bepureunitquaternions,andlet begiven.Setand .Considertherigidmotionof givenby

(2) Let and . Then and are -invariant

orthogonalsubspaceswithrespecttotheinner-product(1),and .Therestrictionofto isarotationthroughtheangle ,andtherestrictionof to isarotation

throughtheangle .Moreover,amatrixrepresentationofthereallinearmapping of

istheblockmatrix

Conversely,anyorientationpreservingrigidmotionof thatfixestheoriginhastheform(2)forsomebasis.

SomedetailsoftheproofofTheorem1aregiveninSection3.Thesubgroup ofthe

orientationpreservingtransformationsin is (3)

Thegroup

€

SO(4) = {A ∈O(4) :Det(A) =1} isisomorphicto .Let bethesetofall2-by-2matriceswithentriesin .Wedenoteby thespaceofcolumnvectors ,with

.Considerthesymplecticscalarproductin givenby

€

〈(v1,v2)T ,(w1,w2)

T 〉 = w 1v1 + w 2v2 . (4) Let begivenandlet

.

Matrixmultiplicationby inducesaright -linearmappingof :

Let bethesetofall suchthat forall .Then

isagroupunderusualmatrixmultiplication:if and ,then

€

PQ =p11q11 + p12q21 p11q12 + p12q22p21q11 + p22q21 p21q12 + p22q22

⎛

⎝ ⎜

⎞

⎠ ⎟ .

Theorderofthemultiplicationoftheentriesin

€

PQ matterssincemultiplicationofquaternionsisnotcommutative.Wesaythat isasymplecticgroup.Welet denotetheconjugatetransposeof .

Lemma2Let begiven.Then ifandonlyif . Proof Let be given. Let and . For , the

symplecticscalarproduct(4)satisfies

Now, the entry of

€

P∗P is . If , then so that. Conversely, suppose . Let and be in . Since

definesaright -linearmappingon ,wehave

Thus, .

Let . Let be the stereographic projection given byif

€

x = x0 + x1i+ x2 j+ x3k ≠ k ,and .Noticethat istherestrictionto oftheinversionin

€

ˆ Η =Η∪{∞} aboutthespherecenteredat andof

radius (page36ofBeardon1983).Let begivenandlet bethereflectionabout

thehyperplanein thatisperpendicularto Set .Then isaMöbius

transformationof .Themapping isexpressedasaquaternioniclinearfractional

transformation as follows. Let where . Set,andconsider

€

m(y) =Y ≡y1 y2−y2 y 1

⎛

⎝ ⎜

⎞

⎠ ⎟ ∈M2(H). (5)

One checks that so that Lemma 2 guarantees that . Moreover,

, so that

€

m y( )( )2 =Y 2 = −I . Weassociateaquaternionic linear fractionaltransformation on by

€

FY (x) = (y1x + y2)(−y2x + y 1)−1,

€

x ∈ ˆ H ∗ (6) (Canlubo and Reyes 2012). Here,we have if , and if

.Moreover,by(Theorem1.2ofCanluboandReyes2012),wehave

€

FY =Π ! fy !Π−1. (7)

Let

€

PSp(2) ≡ Sp(2)/(±I) be a projective symplectic group. Let denote the cosetcontaining . InSection3andintheproofofTheorem3.2 in(CanluboandReyes2012),theauthorsshowthatthecorresponence

€

fy ! [Y ]extendstoawell-definedmapping (8)

that isaninjectivegrouphomomorphism.If ,thenthereexist

€

y(1),…,y(k ) ∈ S3suchthat

€

A = fy(1)!"! f

y(k ) (that is, is a product of reflections). Hence, we have

. In the next section, we shall prove that is differentiable. Byknowingthat isdifferentiable,wewillbeabletoknowtheactionofthedifferential

€

dξ attheLiealgebralevel(seeSection4below).

3Thedifferentiabilityof Theoperatornormofareallinearendomorphism of is

Wederiveaninequalityfortheorientation-preservingrigidmotions inequation(2). Lemma3Let andlet begiven.Supposethat

€

Re(a) = Re(b) . 1.Then isaproductoftworeflections(Coxeter1946). 2.If ,then or . ProofIf

€

Re(a) = Re(b) and ,thenthereexists suchthat (Lemma2.2ofCoxeter1946).Set sothat , ,and

(9) Then isaproductoftworeflections.Since a reflection is its own inverse, the product of two reflections is the identitymapping ifandonly if the tworeflectionsareequal, that is, ifandonly if .Thus,

ifandonlyif ,andthesecondclaimistrueif .

Suppose . The product of two reflections is a rotation in the 2-

dimensional subspace spanned by and . The angle of rotation is twice the anglebetween and ,where

€

cos(θ/2) = Re(zp ) = Re(a) .

Figure1

ConsiderthetriangleinFigure1withsides and .Supposethatthevertexofthetriangle at the intersection of sides and is the origin in . Then the included anglebetween and is ,and bisects .Since

€

fz ! f p isarotation,thelengthofthesideofthetriangleoppositeangle istheoperatornorm

€

Ba,b − I op .Bythecosinelaw,wefind

€

Ba,b − I op2

= 2 1− cosθ( )

€

= 4 1− cos2(θ /2)( )

€

= 2 1−Re(a)( )[ ] 2 1+Re(a)( )[ ]

€

= a −12 a +12.Since ,wehave or .

The results described in Lemma 3 are based on the assumption that

€

Re(a) = Re(b) where .However,fortheproofofTheorem4below,weneedananalysisof

€

Ba,b wherewedonotnecessarilyassume

€

Re(a) = Re(b) .Theanalysisof

€

Ba,b weneedcanbefoundintheproofof(Theorem9.1ofCoxeter1946).Coxetershows

€

Ba,b mayberewrittenasaproductoftworotationsasin(12)wherepossiblyoneoftherotationsistheidentitytransformation.Wedescribe Coxeter’s proof of (12) because the technique used in his proof is applied again inTheorem4.Webeginwiththepolarform(seepage138ofCoxeter1946),andwewrite

where isapureunitquaternion.Since ,wehave

,forevery . (10) Noticethat .DeMoivre’stheoremensuresthat

€

pt = epπ/2( )t = cos(tπ/2) + psin(tπ/2). (11) Thus,weobtain .Similarly,wehave

for somepureunitquaternion and some . Likewise,weobtain . Lemma3guaranteesthat

€

Bpt , q t

isarotationinthereal2-dimensionalsubspace

€

V ={x ∈H : pxq = −x}

through the angle . Notice also that

€

Bpu ,q−u

is a rotation in the subspace

€

W ={x ∈H : pxq = x} through the angle , for any . Choose such thatand .Then

€

Ba,b = Bpt ,q t ! Bpu ,q−u . (12)

Moreover,

€

H =V ⊕W isanorthogonalrealdirectsum;thatis,if

€

x1 ∈V , x2 ∈W ,then

€

Re(x1x 2) = −Re(px1qq x 2p ) = −Re(px1x 2p ) = −Re(x1x 2),and

€

Re(x1x 2) = 0 .Let

€

p1, z1 ∈S3 satisfy

€

z1 = pp1 = p1q .ByLemma3,wehave

€

Bpt ,q t

= f z1 ! f p1 .

Then

€

Bpt ,q t

istheidentitymappingwhenrestrictedto for

€

fz1 ! f p1 (x2) = z1p 1x2p 1z1 = pp1p 1x2p 1z1 = x2q p 1z1 = x2.

Similarly, istheidentitymappingwhenrestrictedto .Thentherestrictionsof

€

Ba,b and

€

Bpt , q t

to

€

V are identical, and the restrictions of

€

Ba,b and

€

Bpt , q t

to

€

W are identical.

Consequently,theoperatornormsatisfies

€

Ba,b − I op =max Bpt , q t − I op, B

pu , q−u− I

op

⎧ ⎨ ⎩

⎫ ⎬ ⎭ . (13)

Wenoteaspecialcasethat

€

Re(a) = Re(b) ifandonlyif or .Wearenow

readytoprovethedifferentiabilityof . Theorem4Thegrouphomomorphism isdifferentiable. ProofSince isagrouphomomorphism,itsufficestoprovecontinuityattheidentity

(Proposition 3.12 of Brocker and Dieck 1985). Note, the subgroup in equation (3) is

isomorphic to , an open subset of that contains . Therefore, it iswithout loss ofgeneralitythatwerestrict to .

Sincethegroupmultiplication in iscontinuous, thereareopenneighborhoodsand of in suchthat

€

V 2 ≡ {v1v2 : v1,v2 ∈V}⊆W . Consider thequotientgroup . If

€

y ∈S3 , let bethecosetdefinedby .Thequotienttopologyin isequivalenttothetopologydefinedbythemetric(page4ofBeckerandKechris1996):

Werecall themapping

€

m : S3 →Sp(2) such that inequation (5).Then

€

m inducesacontinuousmapping suchthat .Sincethecontinuityofgroupmultiplicationonthecompactgroup

€

S3impliesuniformcontinuity,thereexists suchthatwhenever ,and

Now, let satisfy . Applying equation (12), is a product of

tworotations

€

Ba,b = Bpt ,q t ! Bpu ,q−u

forsome ,andpureunitquaternions .Furthermore,eachrotation isaproductoftworeflections.Asinequation(9),thereareunitquaternions satisfying

1.

€

Bpt , q t

= fp tw1! fw1

2.

€

Bpu , q−u

= fpuw2! fw2 .

Notice that

€

Re(pt ) = Re(qt ) by equation (11), and that .Moreover, since,byidentity(13)wehave

Applying Lemma 3 to the above inequality we obtain or . The

metricon satisfies .Sincethenormismultiplicative,wehave

€

d [ptw1],[w1]( ) < δ . Similarly, .Consequently,

€

[m(ptw1)][m(w1)]−1, [m(puw2)][m(w2)]

−1 ∈V . Recall that forall

€

y ∈S3 ,wehave ,so that

€

[m y( )]−1 = [m y( )].Since ,weobtain

€

[m(ptw1)][m(w1)][m(puw2)][m(w2)]∈W . (14)

Ontheotherhand,wealsohave

€

ξ(Ba,b ) = ξ Bpt ,q t ! Bpu ,q−u⎛ ⎝ ⎜ ⎞

⎠ ⎟

€

= ξ fp tw1! fw1 ! f puw2

! fw2⎛ ⎝ ⎜

⎞ ⎠ ⎟

Thus, whenever andhence, iscontinuousattheidentity .

Wecomputethedifferentialof . 4Thedifferential WedescribetheLiealgebra

€

g0oftheLiegroup

€

G0 inequation(3).TheunderlyingsetoftheLiealgebra

€

g0isthesetoftangentvectorsattheidentityof

€

G0 .Astandardprocedureistoconsiderthesurjectivegrouphomomorphism

(15) definedby

€

B(a,b) = Ba,b−1

, whereasbefore, foreach .It isknownthat

the Lie algebra of is the algebra of pure quaternions. Let be the

exponential mapping defined by for every . For , we have (seeequation(10))

and .Now,weevaluatethedifferential of attheidentity.Thecommutative

diagrambelowexplainsthecalculationof thatfollows.

Figure2

Let be given. The tangent vector to the differentiable curve

€

t! etw1 , etw2( ) inat is .Now, isthetangentvectortothecurve

€

t! B etw1, etw2( ) in at Hence,thedifferential

€

dB :H∗ ×H∗ →g0 isaLiealgebrahomomorphismthatsatisfies

€

dB(w1,w2) =ddt t=0

B etw1, etw2( ). In particular, is a real linear mapping of . In fact, the image of under

isdenotedanddefinedby

€

dB(w1,w2)(x) =ddt t=0

etw1xe− t w2[ ] = w1x − xw2. (16) Let

€

gl(H)denotetheLiealgebraofallreallinearmappingsof intoitself.Sincethekernelofis ,thedifferential isaLiealgebraisomorphism.Inparticular,theLiealgebraofis

Moreover, and areisomorphicLiealgebras.

Since whenever isapureunitquaternion,weintroduceanextrafactorinthefollowingcanonicalrealbasisforthevectorspace :

Since isareallinearisomorphism,thefollowingisarealbasisfor : .

Let

€

P : Sp(2)→ PSp(2) be the projectionmapping given by . Since

€

PSp(2) is thequotientgroupof

€

Sp(2)byadiscretegroup

€

±I{ } ,theLiealgebrasof

€

PSp(2)and

€

Sp(2) arethesame.Then and arelocallyisomorphicLiegroups(byTheorem1.11ofHelgason1978).Hence,thereexistsanopenneighborhood of in suchthat

1. isanopensubsetof in , 2.therestriction isadiffeomorphism,and 3. isalocalhomomorphism,thatis,if , , ,then

€

P(x1x2) = P(x1)P(x2). Let denote the inverse function of . Let be the differentiable

homomorphism in Theorem 4. Then is an open subset of . Letbethedifferentiablemappingdefinedby

€

ψ (x) = (PU−1 !ξ)(x)

where .Weevaluate thedifferential of at the elements of . The Lie

algebra of consists of 2-by-2 quaternionic matrices satisfying Forinstance,byequations(16)and(11)wefind

€

dψ dB π2i,0⎛

⎝ ⎞ ⎠

⎛ ⎝ ⎜

⎞ ⎠ ⎟ = dψ

ddt t=0

Bit ,1

⎛

⎝ ⎜

⎞

⎠ ⎟ =

ddt t=0

ψ Bit ,1( )∈ℑ.

Applying equation (12), we express as a product of two rotations. Each rotation is a

productofreflections(seeequation(9)).Then

forsufficientlysmallvaluesoft.Since ,wefind

€

Bit ,1

= PU−1 B

it ,1[ ]( ) forsmallvaluesoft.Since isalocalhomomorphism,weobtain

(17)

However,byapplyingequation(5)wefind

and

€

m(it/2k) =− ksinπ t

4cosπ t

4−cosπ t

4ksin π t

4

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ .

Applyingtheproductruletoequation(17),weobtain

Furthermore,bythechainrule

€

dψ = d PU−1( ) ! dξ.

Thus,weobtainandusetheindicatednotation:

Similarly,byevaluating attheotherelementsof ,wefind

1.

2.

3.

4.

5.

Using the ordered bases and

€

f1, f2, f3, f4, f5, f6{ } for the domain and range of ,respectively,weobtainamatrixrepresentationforthedifferential .

Theorem5Ifwerealize asareallinearisomorphismontoitsimage,thenamatrix

representationfor isthefollowingblockmatrix

Moreover,theeigenvaluesof are ,countingmultiplicities. 5Möbiustransformations In order to describe the higher dimensional analogues of as described in (8), we

describe additional elementary prerequisites. We determine the kernel of the grouphomomorphism in (20) that is inducedby the stereographic projection asdefinedinthissection.ThekerneldescribedinLemma6isneededtocompletetheproofofoursecondmainresult,namely,Theorem11inSection7.

WereviewgeneralfactsaboutMöbiustransformations(seeBeardon1983).Let beagiveninteger.Let betheunitvectorwith1intheithpositionand0elsewheresothat

€

E = {e1,…,en} is thestandardbasis for

€

Rn . Let begiven.Thestandardbilinearformon is

€

K(x,y) =i=1

n

∑xiyi . (18)

Notice that

€

| x | 2=K(x,x). If ,we let

€

x† = | x |−2 x . Let andanonzero begiven.Thehyperplanein containing andperpendicularto is

€

Hn (c,y) = {x ∈Rn :K(x,y) = 0} + c = {x ∈Rn :K(x − c,y) = 0}.

Thereflection

€

τy about isthehomeomorphismof satisfying

€

τy (x) = x − 2K(x − c,y)y†, x ∈Rn,and .

Notice that and that for every The sphere in

centeredat andwithradius isgivenby

€

Sn (c,r) = {x ∈Rn : | x − c |= r}.

Theinversion in isthehomeomorphismof satisfying if and .

Let ,andlet bethestereographicprojectiondefinedby

€

Π(x) = (1− xn )−1 x1,…,xn−1( ), and .

Theinversion in isspecialbecausetherestrictionof to istheinverseof

thestereographicprojection

€

Π,thatis, (19)

(page36ofBeardon1983).Recall,

€

GM( ˆ R n ) isthegroupofMöbius transformationsof

€

ˆ R n ,and is the subgroupofMöbius transformations of that leave theunit sphere

invariant.Thegrouphomomorphism

€

Π∗ :GM(Sn−1) →GM( ˆ R n−1) (20) givenby

€

Π∗( f ) =Π ! f !Π−1,where iswell-definedbecauseofequation(19)andbecausethefiniteproductofMöbiustransformationsisaMöbiustransformation.Wedenoteanyidentitymappingby .

Lemma6Let betheinversionin .Then,thekernelof is .Moreover,

issurjective. ProofIf on ,then istheidentitymappingon .Theonlytwo

Möbius transformations in that restrict to the identity mapping on are the

inversion in andtheidentitymappingon .Thus,thekernelof is . We show that is surjective. Let be given. We apply the Poincaré

extension of satisfying for every (page 34 of

Beardon 1983).We recall the inversion in . The restriction of to is the

stereographicprojection

€

Πthatmaps onto

€

ˆ R n−1.Since

€

φ ext leaves

€

ˆ R n−1invariant,wefind.Applyingequation(19),weobtain

€

Π∗(φ0φextφ0)( ˜ x ) =Πφ0φ

extφ0Π−1( ˜ x ) =Πφ0φ

extφ0φ0( ˜ x ,0) =Πφ0φext ( ˜ x ,0)

since

€

φ02 = I. Moreover,

€

Πφ0φext ( ˜ x ,0) =Πφ0(φ( ˜ x ),0) =φ( ˜ x ) by applying equation (19) again.

Hence, and issurjective. 6VahlenmatricesThe linear fractional transformations on the extended complex plane are well-defined

because any two complex numbers may be multiplied. Similarly, we may define pseudo linear fractional transformations on sincethere isan intrinsicmultiplicationoperation in . Infact, there is a natural embedding of into a real Clifford algebrawhere inherits the

multiplicationof the real Clifford algebra.We identify as in (21).Also, there is a natural(n+1)-dimensional Euclidean space sitting inside the real Clifford algebra as in (23) that

contains .WereturntoourstudyoftheMöbiustransformations of .Wemayextend

toaMöbiustransformationof

€

ˆ V n+1 = Vn+1 ∪{∞}byapplyingthePoincaréextensionof .Then we apply the results in (Ahlfors 1985 and Waterman 1993) to show that is therestrictionto ofsomepseudolinearfractionaltransformationthatisdefinedon

€

ˆ V n+1,seeLemma8. The pseudolinearfractionaltransformationsaredefinedbyVahlenmatrices.

We review general facts about Vahlen matrices (see Ahlfors 1985). Let be therealCliffordalgebrageneratedby subject to and if . For

, consider where . Consider the product

andif ,set .Thecollection

€

{eJ : J ⊆ {1,…,n}}isabasisforthe

realvectorspace .Weidentify withthefollowingsubspaceof :

(21)

Theproductof satisfies

€

xy + yx = −2K(x,y). (22) Let

€

eJ = e j1!e jk be a basis element of . There are three main involutions in. The grade involution is an algebra automorphism of that is denoted and

satisfies

€

ˆ e J = (−1)k e j1!e jk . Theconjugation isanalgebraantiautomorphismof that is

denoted and satisfies

€

e J = (−1)k e jk!e j1 . The reversion is an algebra antiautomorphism of

satisfying

€

eJ∗ = e jk!e j1 . One checks that

€

(eJ∗)∧ = ∗(ˆ e J ) = e J for every

.

For an integer , define the sum

€

s(k) =1+! + k , and set . If , letandset .Welistsomebasicpropertiesof withoutproof.Theproof

ofstatement5followsimmediatelyfrom(Theorem2ofWaterman1993). Lemma7 Let

€

J ⊆ {1,…,n} begiven.Let bethenumberofelementsin .Letbetheproductofallthe for .If ,then

1. ,

2. , 3.

€

exe = −β(n −1)x 4. ,and 5.If and forall ,then .

Let

(23)

If and , themultiplicative inverseof is . Let be the

multiplicativegroupgeneratedbythenonzerovectors in .Wesaythat istheClifford

groupof . AVahlenmatrixisa2-by-2matrix

(24)

where for which , and such that the pseudo-determinant satisfies

€

Det v(A) ≡ ad∗ − bc∗ ∈R −{0} . Let be themultiplicative

group of all Vahlen matrices. If , then

€

Det v(AB) =Det v(A)Det v(B) where thematrix multiplication is the usual one (see equation (32)). Let

€

SL2(Γn ) = {A ∈GL2(Γn ) :Det v(A) =1} .Then isasubgroupof

€

GL2(Γn ) . Let .AVahlenmatrix inducesabijection of onto

itselfdefinedby

€

FA (v) = (av + b)(cv + d)−1, v ∈ ˆ V n+1. (25)

We say that is a pseudo-linear fractional transformation of . It is known that for every . Furthermore, if and only if for some

nonzero . In such a case,wewrite .Notice that is an equivalence relationon

€

GL2(Γn ) . For

€

A ∈GL2(Γn ), let denote the equivalence class of under . Let be a projective group of Vahlen matrices, and let

.

Let denote the group of Möbius transformations of , i.e., the groupgenerated by inversions about n-dimensional spheres, and reflections about hyperplanes in

.Let denotethesubgroupoforientationpreservingMöbiustransformationsof

.For ,themapping definesthegroupisomorphism(TheoremBofAhlfors

1985;Theorem5ofWaterman1993)in

€

PSL2(Γn ) →M( ˆ V n+1). (26)

Notice that for even , the mapping is an orientation reversing orthogonaltransformationof andthemapping definesagroupisomorphism(Ahlfors1985;Waterman1993)in

. (27)

Wecontinue to identify as the subspaceof according to equation (21).We

provethateveryMöbiustransformationof canbeexpressedastherestrictionto ofa

pseudo-linearfractionaltransformation of asin(25). Lemma8Let beaMöbiustransformationof .ThenthereexistsaVahlenmatrix

such that the restriction of the corresponding pseudo linear fractional

transformation satisfies . ProofWeregardthePoincaréextension of asaMöbiustransformationof

satisfying .Foreven , thereexists suchthat becauseof

theisomorphism(27). Forodd , if isorientationpreserving,then forsome

,bytheisomorphism(26);if isorientationreversing,let

€

φ ext2 =φ2 !φext where

is the reflection about in and notice that is an orientation preserving

Möbiustransformationof ,sothat forsome and .

Since , we may extend an orthogonal transformation of to an

orthogonaltransformation of suchthat .Thereareexactlytwoorthogonal

extensions of : with one extension orientation preserving and the other orientation

reversing.Moreover,either or . If with and if is the reflection about the hyperplane , then

€

φ(x) = x − 2K(x,y)y = yxy = FV R n (x) where and . Notice that

is anorientationpreservingorthogonal transformationof by isomorphism (26), but

is orientation reversing. Moreover, by the Cartan-Diedonne theorem, any orthogonal

transformation of (beingaproductof reflections)maybeexpressedas for

some .WedescribeanotherspecificcaseofLemma8.

Lemma 9 Let be the indicated Möbius transformation. There exists

€

A∈GL2(Γn ) thatinducesapseudolinearfractionaltransformation satisfying .

1.If istheinversionin ,then .

2.If isthereflectionabout , ,and ,then .

Proof The proof of the second statement is in the discussion prior to Lemma 9. To

provethefirststatement,let ,andlet betheinversionin .Theinversion

in satisfies

€

φ0 =ψ !φ 1!ψ−1 ,(page23ofBeardon1983).Thus,theproductofthe

Vahlenmatricesfor and istheVahlenmatrix

€

2 en0 1

⎛

⎝ ⎜

⎞

⎠ ⎟ 0 −11 0

⎛

⎝ ⎜

⎞

⎠ ⎟ 1 − en0 2⎛

⎝ ⎜

⎞

⎠ ⎟ =

en −11 −en

⎛

⎝ ⎜

⎞

⎠ ⎟ ,

whichisthematrixinthefirststatement.

7Embeddingtheorthogonalgroup

€

O(n) Let betheinversionin ,andlet

(28)

beasinStatement1ofLemma9. Then .Since ,theinverseoftheVahlen

matrix satisfies and .

Given ,thereexistsaVahlenmatrix

€

A∈GL2(Γn ) suchthat for every (see Lemma 8). Applying the group homomorphism (20) induced bystereographicprojection,andequation(19),weobtain

This leadsus to inner-conjugate theVahlenmatrices inStatement2of Lemma9by . We

present the results without proof. The calculations are straightforward since

Lemma10If

€

i =1,!,n −1,thenthefollowingVahlenmatricesliein :

1.

2. .

Foreach , let denotethereflectionaboutthehyperplane

. We may express as a pseudo-linear fractional transformation. ApplyingLemmata9and10,weobtain where

Now,definethemapping

where . Notice that and that . Let

€

A∗ denote the conjugatetranspose of a Vahlen matrix

€

A = {ai, j}. Here, the (i,j)-entry of

€

A∗ is

€

a j, i . One checks that

,sothat .

For

€

i∈{1,…,k}, let be a unit vector and let . If the product ofreflections satisfy

€

f v1! f vk = In , the identity mapping on , then

€

Π∗( f v1! f vk ) = In−1 .

However, we have

€

Π∗( f v1! f vk ) = FW ˆ R n−1 , where

€

W =V1!Vk =a bc d⎛

⎝ ⎜

⎞

⎠ ⎟ ∈ SL2(Γn−1). Then

for every . Taking , we obtain . Looking at the (1,1)entriesoftheequality ,wehave ;lookingatthe(2,1)entries,wehave ,

sothat ;andlookingatthe(2,2)entries,weget .Now,

€

ad∗ = Det v W( ) =1andwehave .Noticethat becomes ,that is, .

Hence,wehave foreach .Since ,Statement5ofLemma7

guaranteesthat .Consequently, and . Theorem 11 If and if

€

A = f v1! f vk is a product of reflections, then themapping

€

ξen :O(n)→PSL2(Γn−1) (29) givenby

€

ξen (A) = [m1(v1)!m1(vk )]isaninjectivegrouphomomorphism. ProofThediscussionbeforeTheorem11showsthat thegrouphomomorphism(29) is

well-defined. It suffices to show that the group homomorphism (29) is injective. Let

€

A = f v1! f vk beaproductofreflections,let

€

B = f w1! f wl with foreach .Suppose

€

ξen (A) = ξen (B) . Let each . Since

€

ξen (B) = [W1!Wl ], weobtain

sothat

€

FV1!Vk = FW1!Wl and .Recallthatthekernelofthegrouphomomorphism is where is the inversion in (see Lemma 6). Since

€

AB−1 is a real linearmappinginthekernelof ,weconclude .Hence,

€

ξen isinjective. Thehomomorphism (29) is ahigherdimensional analoguebutnotageneralizationof

thehomomorphism

€

ξ in(8).Infact,when

€

n = 4 ,

€

ξen doesnotreduceto

€

ξ asin(8)sincethe

quaternionic matrices in (5) are not necessarily Vahlen matrices. Moreover, the pseudodeterminantof

€

m(y) in(5)isnotnecessarilyarealnumberfor

€

Det v m(y)( ) = p2 + q2 − r2 + y22 + 2rqi − 2prj.WecompareTheorem11withwell-knownfactsstudiedbymanymathstudents.First,if

€

z0 is a complex number with modulus

€

z0 =1, then multiplication by

€

z0 defines a rotation

€

f0(z) = z0z inthecomplexplanethatleavesinvariantthegroup(isomorphicto

€

S1)ofcomplexnumberswithmodulus1.Let

€

z1 ∈S1beasquarerootof

€

z0.Thenthediagonalcomplexmatrix

€

A0 = diag(z1, z1−1) isaVahlenmatrix in

€

SL2(Γ1) thatdefinesa linear fractional transformation

€

FA0 satisfying

€

FA0 = f0 .Next,wemayidentifythesets

€

S 2 (whichisnotagroup)and

€

S 3withthesetofpureunitquaternionsandthegroupofunitquaternions,respectively.If

€

w0 ∈S3isa

unit quaternion, then the mapping

€

g0(w) = w0ww 0, w∈S2 is an orientation preserving

orthogonaltransformationthatleaves

€

S 2invariant.Conversely,anyorthogonaltransformationof

€

R3 (identifiedwith thesetofpurequaternions) that leaves

€

S 2 invariant isof the form

€

g0(Theorem3.3ofCoxeter1946).

In Theorem11, the homomorphism

€

ξen depends on the stereographic projection .Suppose that we change to another stereographic projection.We claim that there is aninjectivegrouphomomorphismfrom intoasubgroupof thatisinner-conjugateto . Notice that by an application of the Poincaré extension, each Möbius

transformation in is the restriction to of some Möbius transformation in. Moreover, for any hyperplane , there is a Möbius transformation in

that maps onto . Let be the group of Möbius

transformations of generated by inversions in spheres in of dimension andreflectionsinhyperplanesof .ThenwemayassociateaMöbiustransformation of toa

Möbius transformation

€

p−1ϕp of by inner-conjugation. A bijection of is aMöbiustransformationifandonlyifthebijectionpreservescrossratios(cf.Theorem3.2.7ofBeardon1983). Thus, eachMöbius transformation in is the restriction to of someMöbius

transformationin .Let ,andlet betheinversioninthesphere .Then mapsthe

unitsphere ontothehyperplane perpendicularto ,thatis,if and ,then

.Therestrictionof

€

ψs totheunitsphere isastereographicprojection;and

€

ψs induces a group homomorphism such that if

€

f ∈GM(Sn−1) we

define . By Lemma7,wehave and

€

ψs = FD ˆ R n where

areVahlenmatriceswithentriesin ,and isthepseudolinearfractionaltransformationof

defined by . Using a proof similar to Statement 1 of Lemma 9, we can show that

.Hence,

€

ψ s∗(FA ) = FDAD −1 ˆ H s .

Let where istheinversioninthesphere .Then isa

Vahlenmatrixforthepseudolinearfractionaltransformationthatdefines .Foreach ,weobtain

Sincethehomomorphism

€

ξen inequation(29)isafunction,therelation

satisfying extends toawell-definedgrouphomomorphism.Thus,by

changingthestereographicprojection

€

Π = φ0 S n-1 =ψen S n-1 toanotherstereographicprojection

,theimagesof

€

ξen and aresubgroupsof thatareinner-conjugatetoeachother.

8LinearmapsonCliffordalgebras Westudy2-by-2matriceswithentries ina realCliffordalgebra .Wediscussan

analogueofrealorthogonalmatrices,andcalltheseCl(n)-orthogonalmatrices.Also,wediscusstheeigenvaluesofsuch2-by-2matriceswithentriesin .

Thebilinearform(18)in extendstotheCliffordalgebra asfollows:

€

KJ∑aJeJ ,

J∑bJeJ

⎛ ⎝ ⎜

⎞ ⎠ ⎟ =

J∑aJbJ

where , and the sum is indexed over all subsets

€

J ⊆ {1,…,n} . The norm satisfies

€

J∑ aJeJ2=

J∑ aJ2 =K

J∑ aJeJ , J∑ aJeJ( ) .Let denotethesetofcolumnvectors where Let beabilinearformon definedby

Let denote the set of all 2-by-2 matrices with entries in . For each

,definearight -linearmapon where

(30)

Thecompositeoftwolinearmaps and isgivenbytheirmatrixproduct

(31)

Then is a group under matrix multiplication. The collection

€

{eJ : J ⊆ {1,…,n}} is anorthonormalbasisfor ,andsatisfies

€

K(eJa,eJb) =K(a,b) =K(aeJ ,beJ ) (32) forall .

Let denotetheconjugatetransposeofthematrix

€

X = {xi, j} .Here,the(i,j)-entryof

€

X ∗is

€

x j, i.Applyingequation(32)andcomponent-wisematrixmultiplication,wefindthat istheadjointof ,thatis,

€

K (2)(Xv,w) =K (2)(v,X ∗w) forall Wesaythat isCl(n)-orthogonalif

€

K (2)(Xv,Xw) =K (2)(v,w) forall .Noticethat isCl(n)-orthogonalifandonlyif(thematrixproducts)

€

XX ∗ = X ∗X = I . Let , ,andlet betheproductofthestandardbasiselementsof

ThenthefollowingthreematricesareCl(n)-orthogonalmatrices:

€

A =v 00 −v⎛

⎝ ⎜

⎞

⎠ ⎟ , B =

0 (−1)n ee 0⎛

⎝ ⎜

⎞

⎠ ⎟ , C =

12(−1)n+1een (−1)

n ee −een

⎛

⎝ ⎜

⎞

⎠ ⎟ .

InadditiontotheVahlenmatrices inLemma9,weclaimthat and inducepseudo linear

fractionaltransformationswhoserestrictionsto aretheinversions and inthespheres

and respectively. Let

€

x ∈Rn be nonzero. Then . ByLemma7,weobtain ,and

€

FB (x) = (−1)n+1β(n)exe/ x 2 = x/ x 2 = φ1(x).

The proof that induces is similar to the proof of Statement 1 of Lemma 9. Let

. Then

€

ψφ 1ψ−1 is the inversion in

€

Sn (en, 2) (see page 23 of Beardon 1983).Consequently,theproductoftheVahlenmatricesfor

€

ψ , φ 1,ψ−1isaVahlenfortheinversionin

€

Sn (en, 2) :

€

2 en0 1

⎛

⎝ ⎜

⎞

⎠ ⎟ 0 (−1)n ee 0⎛

⎝ ⎜

⎞

⎠ ⎟ 1 −en0 2⎛

⎝ ⎜

⎞

⎠ ⎟ =

(−1)n+1een (−1)n e

e −een

⎛

⎝ ⎜

⎞

⎠ ⎟ .

Wemultiplythefactor

€

1/ 2 totheVahlenmatrixintherightsidetoguaranteethat

€

CC∗ = I.We say is positive semidefinite if for every . We write

if ispositivesemidefinite.Forevery ,wehave and .For

thematrixfortheinversionin canbewrittenas

,

and resembles the polar decomposition of real matrices, that is, a product of a positivesemidefinitematrixandanorthogonalmatrixin .

Indiscussingeigenvalues,weusethequadraticnorm , .Notice,itis

notnecessarilytruethat ; if

€

a = e1e2 + e3 then

€

aa = 2(1− e1e2e3) .However, if ,then . We restrict and choose the nonzero eigenvalues and the nonzero

componentsofeigenvectorsfromtheset

€

N∗ ={a∈Cl(n) : 0 ≠ Δ(a)∈R}. Because is themultiplicative group generated by the nonzero vectors in , we have

. It isknownthat

€

Δ(ab) = Δ(a)Δ(b) for (Proposition5.7.2ofGarling2011).Moreover,if then

€

ˆ a ,a∗,a ∈N∗ and . Let

€

X ∈gl 2(n) be given. Suppose that are not both zero. Let be given.We say that is an eigenvector of with right eigenvalue

€

λ if; isaneigenvectorof withlefteigenvalue if .If isaneigenvectorof

€

A with righteigenvalue

€

λ , and if

€

y ∈N∗, thennotice that isalsoaneigenvectorof

€

A withrighteigenvalue

€

yλy −1because

€

A(vy−1) = vλy −1 = vy−1 yλy −1( ). (33) If istheVahlenmatrixin(24),weclaim

€

cv + d∈ Γn ∪{0}forany .Infact,if

€

c ≠ 0,then

€

cv + d = c(v + c −1d)∈ Γn ∪{0} ; while if

€

c = 0, clearly

€

cv + d∈Γn . If the correspondingpseudo linear fractional transformationsatisfies , thenastandardcalculationshowsthat

€

(v,1)T isaneigenvectorof withrighteigenvalue Inparticular,Vahlenmatricesforinversionsandreflectionshaveeigenvectorssincetheyhavefixedpoints.Agooddiscussionoftheeigenvaluesofquaternionicmatricesisin(Zhang2007).

Lemma12Let , ,and with .

1.If isaleftorrighteigenvalueof ,then .

2.Then isalefteigenvalueof ifandonlyif

€

λ = ±v .Also, isaright

eigenvalueof ifandonlyif forsomenonzero

€

b∈N . ProofLet beaneigenvectorof withrighteigenvalue .Then

€

(−1)n r2ev2,ev1( )T= v1λ,v2λ( )

T . Notice that and are each nonzero, otherwise , and is not aneigenvector.Werecallthefunction fromLemma7.Since ,wefind

Then, . Hence, , so that . A similar

calculationshowsthatif isalefteigenvalue,then . Notice that

€

(1,0)T and

€

(0,1)T are eigenvectors of with left eigenvalues and

€

−v,respectively. Also, is neither a left nor a right eigenvalue of . Let be a lefteigenvalueof witheigenvector

€

(x,y)T .Then .If ,thenand ;if ,then and .

Let bearighteigenvalueof correspondingtotheeigenvector ,that

is,suppose .Then . If , then . Either

or .If ,then and .Conversely,ifanonzero

is given, then is a right eigenvalue of corresponding to the eigenvector

, while is a right eigenvalue of corresponding to the eigenvector

.

For the matrix in Lemma 12, suppose is odd, , and . ApplyingStatement4ofLemma7,wefindthat isaleftandarighteigenvalueof witheigenvector

€

(x,1)T .Thus,if isodd, hasinfinitelymanylefteigenvalues;whichisapartialanswertoananalogous question raised for quaternionic matrices in (Question 5.1 of Zhang 2007). InStatement 2 of Lemma 12, the set of right eigenvalues of consists of either one or twodistinct orbits in

€

N under inner-conjugation; we compare this with (Corollary 5.2 of Zhang2007)whereageneraln-by-nquaternionicmatrixhasexactlyndistinctsuchorbits.

9FinalRemarks

We proved that the mapping in(8)isadifferentiablefunctionbetweenLiegroups.Wedeterminedamatrixrepresentationofthedifferential

€

dξ of .InTheorem11,wedescribedahigher dimensional analogue

€

ξen of . Ourmethods involvedMöbius transformations, realClifford algebras, Vahlen matrices, and pseudo linear fractional transformations. We alsodiscussed linearmaps defined by 2-by-2matriceswith entries in a real Clifford algebra, andevaluatedsomeeigenvaluesofsuchmatrices. We conjecture that the homomorphism

€

ξen :O(n)→ PSL2(Γn−1) in Theorem 11 is continuous. Also, it would be interesting to determine the eigenvalues of an arbitrary 2-by-2 matrix with entries in a Clifford algebra. The authors thank the referees for correcting a mistake in one of the matrices in Section 4, and for their suggestions in making this paper more understandable.

References Ahlfors LV. In: Chavel I, Farkas HM. Möbius transformations and Clifford numbers.DifferentialGeometryandComplexAnaysis.Berlin:Springer-Verlag.1985.p.65-73. BeardonAF.1983.TheGeometryofDiscreteGroups.Springer-Verlag.GraduateTextsinMathematics,Vol.91. Becker H, Kechris AS. 1996. The Descriptive Set Theory of Polish Group Actions.

CambridgeUniversityPress.LondonMathematicalSocietyLectureNoteSeries232. Brocker T, Dieck T. 1985. Representations of Compact Lie Groups. Springer-Verlag.GraduateTextsinMathematics,Vol.98. Canlubo C, Reyes E. 2012. Reflections on and quaternionic Möbiustransformations.J.LieTheory.22:839–844. Conway J, Smith D. 2003. On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry. Massachusetts: A.K. Peters, Ltd. CoxeterHSM.1946.Quaternionsandreflections.Amer.Math.Monthly.53:136–146. GarlingDJH.2011.CliffordAlgebras:An Introduction.CambridgeUniversityPress.TheLondonMathematicalSocietyStudentTexts,Vol.78 HelgasonSH.1978,DifferentialGeometry,LieGroups,andSymmetricSpaces.NewYork:AcademicPress. Koecher M, Remmert R. In: H.-D. Ebbinghaus et al. Numbers. New York: Springer; 1991. Graduate Texts in Mathematics, Vol. 123, p.189-220. Lawson J. 2010. Clifford algebras, Möbius transformations, Vahlen matrices, and B-loops.Comment.Math.Univ.Carolin.51:319–331. Waterman PL. 1993. Möbius transformations in several dimensions. Advances inMath.101:87-113. ZhangF.2007.Geršgorintypetheoremsforquaternionicmatrices.LinearAlgebraandItsApplications.424:139–153.